Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions

TL;DR

This paper develops a comprehensive framework for modeling mobile-immobile transport systems with power-law and mixed trapping time distributions, revealing anomalous transport behaviors and concentration profiles relevant across various scientific fields.

Contribution

It unifies multiple modeling approaches and derives explicit forms for anomalous transport dynamics, including power-law decay of mobile mass with Mittag-Leffler trapping times.

Findings

Explicit forms for anomalous transport dynamics

Asymptotic power-law decay of mobile mass

Transient anomalous transport observed even with exponential trapping times

Abstract

We present a framework for systems in which diffusion-advection transport of a tracer substance in a mobile zone is interrupted by trapping in an immobile zone. Our model unifies different model approaches based on distributed-order diffusion equations, exciton diffusion rate models, and random walk models for multi-rate mobile-immobile mass transport. We study various forms for the trapping time dynamics and their effects on the tracer mass in the mobile zone. Moreover we find the associated breakthrough curves, the tracer density at a fixed point in space as function of time, as well as the mobile and immobile concentration profiles and the respective moments of the transport. Specifically we derive explicit forms for the anomalous transport dynamics and an asymptotic power-law decay of the mobile mass for a Mittag-Leffler trapping time distribution. In our analysis we point out that even for exponential trapping time densities transient anomalous transport is observed. Our results have direct applications in geophysical contexts but also in biological, soft matter, and solid state systems.